Fluctuations in Polymers and Random Data Partial Differential Equations

这个项目将研究随机噪声对物理系统理想模型的影响。随机噪声的加入有时会简化复杂系统的分析，但也会从根本上改变它们的行为，并产生意想不到的现象。这两种情况都适用于本项目研究的两类系统：所谓的随机聚合物模型，它描述了凝聚态物质中嘈杂的、不断增长的界面，以及用于模拟量子波和经典波的非线性微分方程。该项目还将更广泛地利用数理统计物理作为动力，对本科生和研究生进行概率论方面的培训。概率论是现代计算机科学和人工智能的重要工具。更具体地说，目标是开发可靠的工具来估计随机聚合物和相关扩散模型的波动，最终目标是将精确代数方法获得的结果扩展到更广泛的模型类别，而精确方法不适用于这些模型。对于非线性波动方程和薛定谔方程，该项目将研究给定随机初始数据的解分布的演变，特别强调小非线性或小色散等极端情况。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project will investigate the effect of random noise on idealized models of physical systems. The addition of random noise sometimes simplifies the analysis of complex systems, but can also fundamentally modify their behavior and give rise to unexpected phenomena. Both situations apply to the two classes of systems studied in this project: so-called random polymer models, which describe noisy, growing interfaces in condensed matter, and nonlinear differential equations that are used to model quantum and classical waves. The project will also use mathematical statistical physics more generally as a motivation to train undergraduate and graduate students in probability theory, an essential tool in modern computer science and artificial intelligence.More specifically, the aim is to develop robust tools to estimate the fluctuations of random polymers and related diffusion models, with the ultimate goal of extending results obtained by exact algebraic methods to broader classes of models where exact methods do not apply. Concerning nonlinear wave and Schroedinger equations, the project will study the evolution of the distribution of solutions for given random initial data, with special emphasis on extremal situations like small nonlinearity or small dispersion.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.